For these purposes a closed cell is a topological space homeomorphic to a simplex, or equally a ball (sphere plus interior) or cube in n dimensions. Only the topological nature matters: but one does want to keep track of the subspace on the 'surface' (the sphere that bounds the ball), and its complement, the interior points. A general cell complex would be a topological space X that is covered by cells; or to put it another way, we start with a space that is the disjoint union of some collection of cells, and take X as a quotient space, for some equivalence relation. This is too general a concept.
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2 CW complexes are defined inductively 3 'The' homotopy category |
A cell is attached by gluing a closed n-dimensional ball Dn to the n-1-skeleton Xn-1, i.e., the union of all lower dimensional cells. The gluing is specified by a continuous function f from ∂Dn = Sn-1 to Xn-1. The points on the new space are exactly the equivalence classes of points in the disjoint union of the old space and the closed cell Dn, the equivalence relation being the transitive closure of x≡f(x). The function f plays an essential role in determining the nature of the newly enlarged complex. For example, if D2 is glued onto S1 in the usual way, we get D2 itself; if f has winding number 2, we get the real projective plane instead.
Assume that X is to be a Hausdorff space: for the purposes of homotopy theory this loses nothing important. Then since closed cells are compact spaces, we can be sure that their images in X are also compact, closed subspaces. From now on, we refer to 'closed cells', and 'open cells', as subspaces of X, the open cell being the image of the distinguished interior.
A 0-cell is just a point; if we only have 0-cells building up a Hausdorff space, it must be a discrete space. The general CW-complex definition can proceed by induction, using this as the base case.
The first restriction is the closure-finite one: each closed cell should be covered by a finite union of open cells.
The other restriction is to do with the possibility of having infinitely many cells, of unbounded dimension. The space X will be presented as a limit of subspaces Xi for i = 0, 1, 2, 3, … . How do we infer a topological structure for X? This is a colimit, in category theory terms. From the continuity of each mapping Xi to X, a closed set in X must have a closed inverse image in each Xi; and so must intersect each closed cell in a closed subset. We can turn this round, and say that a subset C of X is by definition closed precisely when the intersection of C with the closed cells in X is always closed.
With all those preliminaries, the definition of CW-complex runs like this: given X0 a discrete space, and inductively constructed subspaces Xi obtained from Xi-1 by attaching some collection of i-cells, the resulting colimit space X is called a CW-complex provided it is given the weak topology, and the closure-finite condition is satisfied for its closed cells.
The idea of a homotopy category is to start with a topological space category, that is, one in which objects are topological spaces and morphisms are continuous mappings, and abstractly to replace the sets Map(X, Y) of morphisms by sets of equivalence classes Hot(X, Y) that are defined by the homotopy relation. So, the objects remain the same; but the morphisms have been gathered into collections. Under favourable conditions Map(X, Y) is itself a function space and the procedure is to take its set of components under path-connection as a simpler version: this provides the intuitive picture.
The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category. In fact, for technical 'administrative' reasons a homotopy category must keep track of base-points in each space: for example the fundamental group of a connected space is, properly speaking, dependent on the base-point chosen. A base point is in effect a mapping {pt} -> X for each space X; morphisms should respect base points, and all homotopies too. The need to use base points has a significant effect on the products (and other limits) appropriate to use. For example in homotopy theory the smash product of spaces X and Y is used, which is XxY with Xx{y} and {x}xY collapsed to one base point.
To a large extent the business of homotopy theory is to describe the homotopy category; in fact it turns out that calculating Hot(X, Y) is hard, as a general problem, and much effort has been put into the most interesting cases, for example where X and Y are spheres.
Auxiliary constructions may mean that spaces that are not CW complexes must be used on occasion, but half a century since Whitehead has left this definition of homotopy category in good shape. One basic result is that the representable functors on the homotopy category have a simple characterisation (Brown’s representability theorem).
One important later development was that of spectra in homotopy theory, essentially the derived category idea in a form useful for topologists.Attaching cells
CW complexes are defined inductively
'The' homotopy category